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May 2

Biased coin problem

Suppose you have a biased coin but don't know the probability of heads. You flip it n times (n is sufficiently large) but you never get a head. Can you calculate an estimate of the upper bound of the probability of getting a head? Bubba73 ^{You talkin' to me?} 06:14, 2 May 2020 (UTC)[reply]

Probability h = 1 is an upper bound. That did not require much calculation. If you believe in miracles, that is the best you can do. If not, read on. For any value h < 1 the likelihood of the observation is equal to (1−h)ⁿ, which may be small but is not zero. If you set a threshold probability p for what you would consider a miracle (like your pet chimpanzee faultlessly typing out Hamlet's monologue), and seeing as you don't believe in miracles, the inequality (1−h)ⁿ > p should hold, which implies h < 1−p^1/n. You can take the right-hand side as your upper bound. For example, if n = 4489 and p = 10⁻¹⁰⁰, we get h < 0.05. --Lambiam 15:00, 2 May 2020 (UTC)[reply]

That is what I was looking for, except something like p = 10⁻⁹ and the corresponding h being very small. Bubba73 ^{You talkin' to me?} 15:13, 2 May 2020 (UTC)[reply]

Given p and h, you can compute the value of n where you need to start believing in miracles if only tails keeps coming up: n > log_1−h p = (log p)/log(1−h). For p = 10⁻⁹ and h = 0.001, this comes out as n > 20712. A good approximation for very small h is 20.723266 × (1/h − 1/2). --Lambiam 17:37, 2 May 2020 (UTC)[reply]

The key concept here is Statistical significance. --RDBury (talk) 20:37, 2 May 2020 (UTC)[reply]

What would be the null hypothesis? --Lambiam 21:08, 2 May 2020 (UTC)[reply]

This is related the sunrise problem (see also rule of succession) and the answer depends on the prior probability distribution you assign to the coin's bias before you start flipping. You are then asking about the conditional distribution where you update the prior on seeing n heads. A uniform prior might not be realistic. 2602:24A:DE47:B270:DDD2:63E0:FE3B:596C (talk) 20:01, 3 May 2020 (UTC)[reply]

These approaches give (debatable) estimates, but no upper bounds. --Lambiam 08:14, 4 May 2020 (UTC)[reply]

If you don't know anything about the prior, for any n, there is (edited:) no upper bound less than 1. p could be almost 1, and you might have flipped a very unlikely string of tails. 2602:24A:DE47:B270:DDD2:63E0:FE3B:596C (talk) 08:23, 5 May 2020 (UTC) (typo fixed, 09:18, 6 May 2020 (UTC))[reply]